1. The problem is to analyze the function $$f(x) = \frac{x^3 + x^2 - 4x - 4}{x^3 - x^2 - 8x + 12}$$ and then construct its graph. 2. First, factor numerator and denominator to simplify and find key features. 3. Factor numerator: $$x^3 + x^2 - 4x - 4$$ Group terms: $$ (x^3 + x^2) - (4x + 4) = x^2(x + 1) - 4(x + 1) = (x + 1)(x^2 - 4)$$ Further factor: $$x^2 - 4 = (x - 2)(x + 2)$$ So numerator factors as $$ (x + 1)(x - 2)(x + 2)$$ 4. Factor denominator: $$x^3 - x^2 - 8x + 12$$ Group terms: $$ (x^3 - x^2) - (8x - 12) = x^2(x - 1) - 4(2x - 3)$$ Try to factor by grouping or find roots. Check possible roots using Rational Root Theorem: try $x=1$: $$1 - 1 - 8 + 12 = 4 \neq 0$$ Try $x=2$: $$8 - 4 - 16 + 12 = 0$$ So $x=2$ is a root. Divide denominator by $(x - 2)$: Using synthetic division: Coefficients: 1, -1, -8, 12 Bring down 1 Multiply 1*2=2, add to -1 = 1 Multiply 1*2=2, add to -8 = -6 Multiply -6*2 = -12, add to 12 = 0 So quotient is $$x^2 + x - 6$$ Factor quadratic: $$x^2 + x - 6 = (x + 3)(x - 2)$$ 5. So denominator factors as $$ (x - 2)(x + 3)(x - 2) = (x - 2)^2 (x + 3)$$ 6. Simplify the function: $$f(x) = \frac{(x + 1)(x - 2)(x + 2)}{(x - 2)^2 (x + 3)} = \frac{(x + 1)(x + 2)}{(x - 2)(x + 3)}$$ for $x \neq 2$. 7. Identify vertical asymptotes where denominator is zero and numerator is not zero: - At $x = 2$, denominator zero but numerator is $(2 + 1)(2 + 2) = 3 \times 4 = 12 \neq 0$, so vertical asymptote at $x=2$. - At $x = -3$, denominator zero and numerator is $(-3 + 1)(-3 + 2) = (-2)(-1) = 2 \neq 0$, so vertical asymptote at $x = -3$. 8. Find horizontal asymptote by comparing degrees: Both numerator and denominator are degree 2 after simplification. Leading coefficients numerator: $x \times x = 1$. Leading coefficients denominator: $x \times x = 1$. So horizontal asymptote is $y = \frac{1}{1} = 1$. 9. Find x-intercepts by setting numerator zero: $$(x + 1)(x + 2) = 0 \Rightarrow x = -1, -2$$ 10. Find y-intercept by evaluating $f(0)$: $$f(0) = \frac{(0 + 1)(0 + 2)}{(0 - 2)(0 + 3)} = \frac{1 \times 2}{-2 \times 3} = \frac{2}{-6} = -\frac{1}{3}$$ 11. Sketch the graph using vertical asymptotes at $x = 2$ and $x = -3$, horizontal asymptote at $y=1$, x-intercepts at $x = -1$ and $x = -2$, and y-intercept at $y = -\frac{1}{3}$. Final answer: The simplified function is $$f(x) = \frac{(x + 1)(x + 2)}{(x - 2)(x + 3)}$$ with vertical asymptotes at $x=2$ and $x=-3$, horizontal asymptote at $y=1$, x-intercepts at $x=-1$ and $x=-2$, and y-intercept at $y=-\frac{1}{3}$.